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Theorem icocncflimc 37761
Description: Limit at the lower bound, of a continuous function defined on a left closed right open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
icocncflimc.a  |-  ( ph  ->  A  e.  RR )
icocncflimc.b  |-  ( ph  ->  B  e.  RR* )
icocncflimc.altb  |-  ( ph  ->  A  <  B )
icocncflimc.f  |-  ( ph  ->  F  e.  ( ( A [,) B )
-cn-> CC ) )
Assertion
Ref Expression
icocncflimc  |-  ( ph  ->  ( F `  A
)  e.  ( ( F  |`  ( A (,) B ) ) lim CC  A ) )

Proof of Theorem icocncflimc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 icocncflimc.f . . 3  |-  ( ph  ->  F  e.  ( ( A [,) B )
-cn-> CC ) )
2 icocncflimc.a . . . . 5  |-  ( ph  ->  A  e.  RR )
32rexrd 9687 . . . 4  |-  ( ph  ->  A  e.  RR* )
4 icocncflimc.b . . . 4  |-  ( ph  ->  B  e.  RR* )
52leidd 10177 . . . 4  |-  ( ph  ->  A  <_  A )
6 icocncflimc.altb . . . 4  |-  ( ph  ->  A  <  B )
73, 4, 3, 5, 6elicod 11682 . . 3  |-  ( ph  ->  A  e.  ( A [,) B ) )
81, 7cnlimci 22837 . 2  |-  ( ph  ->  ( F `  A
)  e.  ( F lim
CC  A ) )
9 cncfrss 21916 . . . . . . . 8  |-  ( F  e.  ( ( A [,) B ) -cn-> CC )  ->  ( A [,) B )  C_  CC )
101, 9syl 17 . . . . . . 7  |-  ( ph  ->  ( A [,) B
)  C_  CC )
11 ssid 3450 . . . . . . 7  |-  CC  C_  CC
12 eqid 2450 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
13 eqid 2450 . . . . . . . 8  |-  ( (
TopOpen ` fld )t  ( A [,) B
) )  =  ( ( TopOpen ` fld )t  ( A [,) B ) )
14 eqid 2450 . . . . . . . 8  |-  ( (
TopOpen ` fld )t  CC )  =  ( ( TopOpen ` fld )t  CC )
1512, 13, 14cncfcn 21934 . . . . . . 7  |-  ( ( ( A [,) B
)  C_  CC  /\  CC  C_  CC )  ->  (
( A [,) B
) -cn-> CC )  =  ( ( ( TopOpen ` fld )t  ( A [,) B ) )  Cn  ( ( TopOpen ` fld )t  CC ) ) )
1610, 11, 15sylancl 667 . . . . . 6  |-  ( ph  ->  ( ( A [,) B ) -cn-> CC )  =  ( ( (
TopOpen ` fld )t  ( A [,) B
) )  Cn  (
( TopOpen ` fld )t  CC ) ) )
171, 16eleqtrd 2530 . . . . 5  |-  ( ph  ->  F  e.  ( ( ( TopOpen ` fld )t  ( A [,) B ) )  Cn  ( ( TopOpen ` fld )t  CC ) ) )
1812cnfldtopon 21796 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
1918a1i 11 . . . . . . 7  |-  ( ph  ->  ( TopOpen ` fld )  e.  (TopOn `  CC ) )
20 resttopon 20170 . . . . . . 7  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ( A [,) B )  C_  CC )  ->  ( (
TopOpen ` fld )t  ( A [,) B
) )  e.  (TopOn `  ( A [,) B
) ) )
2119, 10, 20syl2anc 666 . . . . . 6  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A [,) B ) )  e.  (TopOn `  ( A [,) B ) ) )
2212cnfldtop 21797 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  Top
23 unicntop 37365 . . . . . . . . 9  |-  CC  =  U. ( TopOpen ` fld )
2423restid 15325 . . . . . . . 8  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
2522, 24ax-mp 5 . . . . . . 7  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
2625cnfldtopon 21796 . . . . . 6  |-  ( (
TopOpen ` fld )t  CC )  e.  (TopOn `  CC )
27 cncnp 20289 . . . . . 6  |-  ( ( ( ( TopOpen ` fld )t  ( A [,) B ) )  e.  (TopOn `  ( A [,) B ) )  /\  ( ( TopOpen ` fld )t  CC )  e.  (TopOn `  CC ) )  -> 
( F  e.  ( ( ( TopOpen ` fld )t  ( A [,) B ) )  Cn  ( ( TopOpen ` fld )t  CC ) )  <->  ( F : ( A [,) B ) --> CC  /\  A. x  e.  ( A [,) B ) F  e.  ( ( ( ( TopOpen ` fld )t  ( A [,) B ) )  CnP  ( ( TopOpen ` fld )t  CC ) ) `  x ) ) ) )
2821, 26, 27sylancl 667 . . . . 5  |-  ( ph  ->  ( F  e.  ( ( ( TopOpen ` fld )t  ( A [,) B ) )  Cn  ( ( TopOpen ` fld )t  CC ) )  <->  ( F : ( A [,) B ) --> CC  /\  A. x  e.  ( A [,) B ) F  e.  ( ( ( ( TopOpen ` fld )t  ( A [,) B ) )  CnP  ( ( TopOpen ` fld )t  CC ) ) `  x ) ) ) )
2917, 28mpbid 214 . . . 4  |-  ( ph  ->  ( F : ( A [,) B ) --> CC  /\  A. x  e.  ( A [,) B
) F  e.  ( ( ( ( TopOpen ` fld )t  ( A [,) B ) )  CnP  ( ( TopOpen ` fld )t  CC ) ) `  x
) ) )
3029simpld 461 . . 3  |-  ( ph  ->  F : ( A [,) B ) --> CC )
31 ioossico 11720 . . . 4  |-  ( A (,) B )  C_  ( A [,) B )
3231a1i 11 . . 3  |-  ( ph  ->  ( A (,) B
)  C_  ( A [,) B ) )
33 eqid 2450 . . 3  |-  ( (
TopOpen ` fld )t  ( ( A [,) B )  u.  { A } ) )  =  ( ( TopOpen ` fld )t  ( ( A [,) B )  u. 
{ A } ) )
342recnd 9666 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
3523ntrtop 20079 . . . . . . . . 9  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( int `  ( TopOpen
` fld
) ) `  CC )  =  CC )
3622, 35ax-mp 5 . . . . . . . 8  |-  ( ( int `  ( TopOpen ` fld )
) `  CC )  =  CC
37 undif 3847 . . . . . . . . . . 11  |-  ( ( A [,) B ) 
C_  CC  <->  ( ( A [,) B )  u.  ( CC  \  ( A [,) B ) ) )  =  CC )
3810, 37sylib 200 . . . . . . . . . 10  |-  ( ph  ->  ( ( A [,) B )  u.  ( CC  \  ( A [,) B ) ) )  =  CC )
3938eqcomd 2456 . . . . . . . . 9  |-  ( ph  ->  CC  =  ( ( A [,) B )  u.  ( CC  \ 
( A [,) B
) ) ) )
4039fveq2d 5867 . . . . . . . 8  |-  ( ph  ->  ( ( int `  ( TopOpen
` fld
) ) `  CC )  =  ( ( int `  ( TopOpen ` fld ) ) `  (
( A [,) B
)  u.  ( CC 
\  ( A [,) B ) ) ) ) )
4136, 40syl5eqr 2498 . . . . . . 7  |-  ( ph  ->  CC  =  ( ( int `  ( TopOpen ` fld )
) `  ( ( A [,) B )  u.  ( CC  \  ( A [,) B ) ) ) ) )
4234, 41eleqtrd 2530 . . . . . 6  |-  ( ph  ->  A  e.  ( ( int `  ( TopOpen ` fld )
) `  ( ( A [,) B )  u.  ( CC  \  ( A [,) B ) ) ) ) )
4342, 7elind 3617 . . . . 5  |-  ( ph  ->  A  e.  ( ( ( int `  ( TopOpen
` fld
) ) `  (
( A [,) B
)  u.  ( CC 
\  ( A [,) B ) ) ) )  i^i  ( A [,) B ) ) )
4422a1i 11 . . . . . 6  |-  ( ph  ->  ( TopOpen ` fld )  e.  Top )
45 ssid 3450 . . . . . . 7  |-  ( A [,) B )  C_  ( A [,) B )
4645a1i 11 . . . . . 6  |-  ( ph  ->  ( A [,) B
)  C_  ( A [,) B ) )
4723, 13restntr 20191 . . . . . 6  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  ( A [,) B
)  C_  CC  /\  ( A [,) B )  C_  ( A [,) B ) )  ->  ( ( int `  ( ( TopOpen ` fld )t  ( A [,) B ) ) ) `  ( A [,) B ) )  =  ( ( ( int `  ( TopOpen ` fld )
) `  ( ( A [,) B )  u.  ( CC  \  ( A [,) B ) ) ) )  i^i  ( A [,) B ) ) )
4844, 10, 46, 47syl3anc 1267 . . . . 5  |-  ( ph  ->  ( ( int `  (
( TopOpen ` fld )t  ( A [,) B ) ) ) `
 ( A [,) B ) )  =  ( ( ( int `  ( TopOpen ` fld ) ) `  (
( A [,) B
)  u.  ( CC 
\  ( A [,) B ) ) ) )  i^i  ( A [,) B ) ) )
4943, 48eleqtrrd 2531 . . . 4  |-  ( ph  ->  A  e.  ( ( int `  ( (
TopOpen ` fld )t  ( A [,) B
) ) ) `  ( A [,) B ) ) )
507snssd 4116 . . . . . . . . 9  |-  ( ph  ->  { A }  C_  ( A [,) B ) )
51 ssequn2 3606 . . . . . . . . 9  |-  ( { A }  C_  ( A [,) B )  <->  ( ( A [,) B )  u. 
{ A } )  =  ( A [,) B ) )
5250, 51sylib 200 . . . . . . . 8  |-  ( ph  ->  ( ( A [,) B )  u.  { A } )  =  ( A [,) B ) )
5352eqcomd 2456 . . . . . . 7  |-  ( ph  ->  ( A [,) B
)  =  ( ( A [,) B )  u.  { A }
) )
5453oveq2d 6304 . . . . . 6  |-  ( ph  ->  ( ( TopOpen ` fld )t  ( A [,) B ) )  =  ( ( TopOpen ` fld )t  ( ( A [,) B )  u. 
{ A } ) ) )
5554fveq2d 5867 . . . . 5  |-  ( ph  ->  ( int `  (
( TopOpen ` fld )t  ( A [,) B ) ) )  =  ( int `  (
( TopOpen ` fld )t  ( ( A [,) B )  u. 
{ A } ) ) ) )
56 snunioo1 37607 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  (
( A (,) B
)  u.  { A } )  =  ( A [,) B ) )
573, 4, 6, 56syl3anc 1267 . . . . . 6  |-  ( ph  ->  ( ( A (,) B )  u.  { A } )  =  ( A [,) B ) )
5857eqcomd 2456 . . . . 5  |-  ( ph  ->  ( A [,) B
)  =  ( ( A (,) B )  u.  { A }
) )
5955, 58fveq12d 5869 . . . 4  |-  ( ph  ->  ( ( int `  (
( TopOpen ` fld )t  ( A [,) B ) ) ) `
 ( A [,) B ) )  =  ( ( int `  (
( TopOpen ` fld )t  ( ( A [,) B )  u. 
{ A } ) ) ) `  (
( A (,) B
)  u.  { A } ) ) )
6049, 59eleqtrd 2530 . . 3  |-  ( ph  ->  A  e.  ( ( int `  ( (
TopOpen ` fld )t  ( ( A [,) B )  u.  { A } ) ) ) `
 ( ( A (,) B )  u. 
{ A } ) ) )
6130, 32, 10, 12, 33, 60limcres 22834 . 2  |-  ( ph  ->  ( ( F  |`  ( A (,) B ) ) lim CC  A )  =  ( F lim CC  A ) )
628, 61eleqtrrd 2531 1  |-  ( ph  ->  ( F `  A
)  e.  ( ( F  |`  ( A (,) B ) ) lim CC  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886   A.wral 2736    \ cdif 3400    u. cun 3401    i^i cin 3402    C_ wss 3403   {csn 3967   class class class wbr 4401    |` cres 4835   -->wf 5577   ` cfv 5581  (class class class)co 6288   CCcc 9534   RRcr 9535   RR*cxr 9671    < clt 9672   (,)cioo 11632   [,)cico 11634   ↾t crest 15312   TopOpenctopn 15313  ℂfldccnfld 18963   Topctop 19910  TopOnctopon 19911   intcnt 20025    Cn ccn 20233    CnP ccnp 20234   -cn->ccncf 21901   lim CC climc 22810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fi 7922  df-sup 7953  df-inf 7954  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ico 11638  df-icc 11639  df-fz 11782  df-seq 12211  df-exp 12270  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-plusg 15196  df-mulr 15197  df-starv 15198  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-rest 15314  df-topn 15315  df-topgen 15335  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-ntr 20028  df-cn 20236  df-cnp 20237  df-xms 21328  df-ms 21329  df-cncf 21903  df-limc 22814
This theorem is referenced by:  fourierdlem46  38010
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